7.4 QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS *

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1 Andri Tokmakoff, MIT Dparmn of Chmisry, 5/19/ QUANTUM MECANICAL TREATMENT OF FLUCTUATIONS * Inroducion and Prviw Now h origin of frquncy flucuaions is inracions of our molcul (or mor approprialy our lcronic ransiions) wih is nvironmn. This should hav a clos rlaionship o our Displacd armonic Oscillaor modl, which is a gnral approach of a ransiion coupld o vibraions. For ha modl w found ha i µ µ = µ µ i pn n n n = µ i i g i In h cas of im-dpndn changs o h lcronic nrgy gap, w can imagin wriing h sam problm in rms of a amilonian ha dscribs h lcronic nrgy gap s dpndnc on Q (dviaion rlaiv o ): = = (Enrgy Gap amilonian) E G g Wha w will show is ha whn h im-dpndnc of h sysm is xprssd in rms of i Cµµ xp+ d i = µ () This xprssion is consisn wih h obsrvaion of a random flucuaion of h ransiion nrgy if w qua ( ) ( ) So, w should xpc ha coupling of an lcronic ransiion o a coninuous disribuion of harmonic oscillaors (a bah) should b mahmaically quivaln o a sochasic modulaion of h nrgy gap. * S Mukaml, Ch. 8 and Ch. 7

2 7-1 This bins o mak a lil mor sns if w look a lil closr a h amilonian: = + E + g + E g = + + g g = p m + 1 m D Q g = = m Q d m Q = m d Q + m d linar in Q No ha his looks idnical o a amilonian ha dscribs h coupling of an lcronic sysm o a bah [on dr of frdom hr] of.o. wih a linar coupling bwn h wo = + + S B SB = E + + g E g S g p 1 B = + m Q m SB = m d Q coupling srngh r w no h similariy SB. 5/19/5

3 7-13 ENERGY GAP AMILTONIAN Now l s work hrough his mor carfully. W will sar b r-wriing our amilonian in rms of rducd coordinas = + E + + E g g = + + g whr = g m p = p q = d = m m q d Slop (coupling) = λ/d = p + q d ( ) g = p + q = g = d q + d Now, h absorpion linshap is dscribd hrough µ ( )µ : i = µ µ = µ C F µµ i g i F = 5/19/5

4 7-14 If w wan o rwri his in rms of, w ar changing h rprsnaion of h dynamics o a nw amilonian. Similar o h ransformaion o h inracion picur, w will choos o rprsn h im dpndnc of by voluion undr g = + = + V g i U i g i = xp = U U g + d ) ( ( ) = i g i g = U U g g This implis: i g i i F ( ) = = xp d ( ) ) + ( No: Transformaion o a nw amilonian If w hav i A A i B and w wan o xprss his in rms of A i ( B A ) = A i BA w will now b volving h sysm undr a diffrn amilonian BA, and w mus prform a ransformaion ino his nw fram of rfrnc. In gnral if you wan o chang o a nw amilonian, you nd o prform a uniary ransformaion undr h rfrnc amilonian: nw = rf + diff 5/19/5

5 7-15 i i i nw rf = xp+ d diff ) ( ( ) = U U diff rf diff rf This is wha w did for h inracion picur. Now, procding a bi diffrnly, w can xprss h im voluion undr h amilonian of B rlaiv o A as = + B A BA i ib i A = xp+ d BA ) ( + i A A whr ( ) BA i = BA. This implis: i + i A ib = xp+ d BA ) ( Th cumulan xpansion o scond ordr says: i i F ( ) = xp d () + d d 1 ( ) ( 1) + ) * ( - +, (. i i = xp / + ) * d d 1 C (, 1) + ( No im ordring now. - +, (. r, w nod ha h way w dfind mans ha d = = (Th nrgy gap could also b dfind rlaiv o h nrgy gap a Q = : = g.) Also, w dfind ( ) = ( ) ( ) ( ) ( ) C, = 1 1 = 5/19/5

6 7-16 So w hav C µµ ( ) = µ ( g ) i E E + / g = ( ) g d d 1 1 This is h corrlaion funcion xprssion ha drmins h absorpion linshap for a imdpndn nrgy gap. Now, valuaing C ( ) ( ) ( = for on harmonic oscillaor = C p n n n n 1 i g i g = p ( n n n n + ( 1) = + + = i i D n n D d Again n is h hrmally avragd occupaion numbr. Also = ( )( 1 ) + ( ) g D coh / cos i sin coh x = x x + x x = g( + ig(( No w now hav ral ( g )and imaginary g Alrnaivly, w can wri his as i + i i ( 1 1) ( 1) i + i ( 1)( 1) ( 1) conribuions o F( ). g = D n + + id = D n n + + i D Vibraional progrssion o bands / 5/19/5

7 7-17 A low T, coh ( / ) 1 and n = [ 1 + ] g D cos i sin i i = D 1 i Combining wih F = id g W hav our old rsul: i = ( 1) F xp D 5/19/5

8 7-18 Disribuion of Nuclar Sas Coupling o a disribuion of sas characrizd by a dnsiy of sas W ( D ). As discussd bfor, w xpc = ( ) ( ) F xp d W g, Coupling o a coninuum will induc irrvrsibl rlaxaion, which will b characrizd by damping of C. This is achivd by summing ovr a disribuion of C (,) : = ( ) ( ) C d C, W ( ) Alrnaivly in h frquncy domain: + i C () = * C ( ) d + i * d W * C, d C ( ) = C D n n ( ) = ( ) ( + 1) ( ) + ( + ) ( D C)) = + + ( W dfin a spcral dnsiy or coupling-wighd dnsiy of sas: = = d W D = W D C This lads o: 5/19/5

9 7-19 C () + 1 g ( ) = 4 d xp ( i ) + i 1 ( + * ) + = 4 d,() coh ( 1 cos ) + i( sin ) / = d, This is a prfcly gnral xprssion for h linshap funcion in rms of an arbirary spcral disribuion dscribing h im-scal and ampliud of nrgy gap flucuaions. Givn a spcral dnsiy ρ(ω), you can calcula spcroscopy and ohr im-dpndn procsss in a flucuaing nvironmn. Spcial cas: Now ak h spcific cas ha w choos a Lornzian spcral dnsiy cnrd a ω=: C ( ) = + In h high mpraur limi kt >> w g: [ ] g( ) = kt xp () + 1 i [ xp () + 1 ] So if w ignor h imaginary par of g( ), and w qua = kt w hav our sochasic modl: g( ) = c xp / c c = 1 [ + / c 1] 5/19/5

10 7- So, h inracion of an lcronic ransiion wih a frquncy disribuion of nuclar coordinas (a bah) lads o lin broadning and irrvrsibl rlaxaion. Th ffc is o damp h nuclar oscillaions on lcronic sas. Mor commonly w would hink of our lcronic ransiion coupld o a paricular nuclar coordina Q which may b a local mod, bu h local mod fls a flucuaing nvironmn a fricion. Classically, w would undrsand h flucuaions as Brownian moion, dscribd by a gnralizd Langvin quaion: + + ( ) ( ) = + mq m Q m d Q f F.O. damping, for no mmory m Q random forc For a random forc: f ( ) = For no mmory: ( ) = ( ) f ( ) f () = mkt ( ) This oscillaor has a corrlaion funcion dscribd by C QQ 1 + i Looks similar o a dampd.o. This coordina corrlaion funcion is jus wha w nd for dscribing linshaps. No: = = C d q q W can g xacly h sam bhavior as h classical GLE by coupling o a bah of harmonic oscillaors (normal mods, x ). For 5/19/5

11 7-1 N hnuc = p x ) + whr x q ( = 1 Wih his amilonian, w can consruc N harmonic coordinas any way w lik wih h appropria uniary ransformaion. W wan o ransform o our local mod Q: U x = Q X 1 X X n1 Now: N 1 * * = 1 sysm bah sysm-bah inracion hnuc = p Q + + p + X + Q c X So, going back o our displacd.o. problm, w can rwri our amilonian o includ h inracion of on primary vibraion wih a bah, which lads o damping: Elcronic ransiion Primary vibraion Bah of.o.s 5/19/5

12 7- Brownian Oscillaor amilonian (Spin-Boson amilonian) Th formulaion of h nrgy gap amilonian in which w assum ha h lcronic disribuion is coupld o nuclar coordinas dscribd by a disribuion of harmonic oscillaors is known as h Brownian oscillaor modl (popularizd by Mukaml). I is obaind as follows: = + + s B SB = + g g TOT TOT S g B = p x ) + ( = q c x c : coupling SB ) = C = q q r = d is h masur of h coupling of our primary oscillaor o h lcronic ransiion. Th corrlaion funcions for q ar complicad o solv for, bu can b don analyically: C () = m ( ) + () whr γ is h spcral disribuion of couplings bwn our primary vibraion and h bah = c For a consan, () : 1 C ( ) = xp( / ) sin m = / 4 rducd frquncy 5/19/5

13 7-3 This modl inrpolas bwn h cohrn undampd limi and h ovrdampd sochasic limi. If w s, w rcovr our arlir rsul for C () and g() for coupling o undampd nuclar coordinas. For wak damping << 1 C ( ) xp ( / ) sin For srong damping >> i, is imaginary and C ( ) xp( ) = D which is h sochasic modl. Absorpion linshaps ar calculad as bfor, by calculaing h linshap funcion from h spcral dnsiy abov. This modl allows a bah o b consrucd wih all possibl im scals, by summing ovr many nuclar drs of frdom, ach of which may b undr- or ovr-dampd. C () = C,i () = i. m i i ( ) i + i () 5/19/5